# A higher-dimensional Contou-Carrère symbol: local theory

@article{Gorchinskiy2015AHC, title={A higher-dimensional Contou-Carr{\`e}re symbol: local theory}, author={S. Gorchinskiy and Denis Vasilievich Osipov}, journal={Sbornik: Mathematics}, year={2015}, volume={206}, pages={1191 - 1259} }

We construct a higher-dimensional Contou-Carrère symbol and we study some of its fundamental properties. The higher-dimensional Contou-Carrère symbol is defined by means of the boundary map for -groups. We prove its universal property. We provide an explicit formula for the higher-dimensional Contou-Carrère symbol over and we prove the integrality of this formula. We also study its relation with the higher-dimensional Witt pairing. Bibliography: 46 titles.

## 16 Citations

### Higher-dimensional Contou-Carrère symbol and continuous automorphisms

- MathematicsFunctional Analysis and Its Applications
- 2016

We prove that the higher-dimensional Contou-Carrère symbol is invariant under the continuous automorphisms of algebras of iterated Laurent series over a ring. Applying this property, we obtain a new…

### Higher-dimensional Contou-Carrère symbol and continuous automorphisms

- Mathematics
- 2016

We prove that the higher-dimensional Contou-Carrère symbol is invariant under the continuous automorphisms of algebras of iterated Laurent series over a ring. Applying this property, we obtain a new…

### Second Chern numbers of vector bundles and higher adeles

- Mathematics
- 2017

We give a construction of the second Chern number of a vector bundle over a smooth projective surface by means of adelic transition matrices for the vector bundle. The construction does not use an…

### Tangent space to Milnor K-groups of rings

- Mathematics
- 2015

We prove that the tangent space to the (n + 1)th Milnor K-group of a ring R is isomorphic to the group of nth absolute Kähler differentials of R when the ring R contains 1/2 and has sufficiently many…

### On the Calabi–Yau Compactifications of Toric Landau–Ginzburg Models for Fano Complete Intersections

- MathematicsMathematical Notes
- 2018

It is well known that Givental’s toric Landau–Ginzburg models for Fano complete intersections admit Calabi–Yau compactifications. We give an alternative proof of this fact. As a consequence of this…

### Laurent phenomenon for Landau-Ginzburg models of complete intersections in Grassmannians

- Mathematics
- 2014

In 1997 Batyrev, Ciocan-Fontanine, Kim, and van Straten suggested a construction of Landau–Ginzburg models for Fano complete intersections in Grassmannians similar to Givental’s construction for…

### Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds

- Mathematics
- 2016

We prove that smooth Fano threefolds have toric Landau- Ginzburg models. More precisely, we prove that their Landau-Ginzburg models, represented as Laurent polynomials, admit compactifications to…

### Formal Bott-Thurston cocycle and part of formal Riemann-Roch theorem

- Mathematics
- 2022

The Bott-Thurston cocycle is a 2 -cocycle on the group of orientation-preserving diﬀeomorphisms of the circle. We introduce and study the formal analog of Bott-Thurston cocycle. The formal…

### Relative Milnor -groups and differential forms of split nilpotent extensions

- MathematicsIzvestiya: Mathematics
- 2018

Let be a commutative ring and a nilpotent ideal such that the quotient splits out of as a ring. Let be an integer such that . We establish a canonical isomorphism between the relative Milnor -group…

### Continuous homomorphisms between algebras of iterated Laurent series over a ring

- Mathematics
- 2016

We study continuous homomorphisms between algebras of iterated Laurent series over a commutative ring. We give a full description of such homomorphisms in terms of discrete data determined by the…

## References

SHOWING 1-10 OF 51 REFERENCES

### The two-dimensional Contou-Carrère symbol and reciprocity laws

- Mathematics
- 2013

We define a two-dimensional Contou-Carrere symbol, which is a deformation of the two-dimensional tame symbol and is a natural generalization of the (usual) one-dimensional Contou-Carrere symbol. We…

### Explicit formula for the higher-dimensional Contou-Carrère symbol

- Physics, Mathematics
- 2015

In this note we give an explicit formula for the n-dimensional Contou-Carrère symbol (see Theorem 1) in the case when the ground ring contains the field Q. Throughout this note let A be an arbitrary…

### The index map in algebraic K-theory

- Mathematics
- 2014

In this paper we provide a detailed description of the K-theory torsor constructed by S. Saito for a Tate R-module, and its analogue for general idempotent complete exact categories. We study the…

### To the multidimensional tame symbol

- Mathematics
- 2003

We give a construction of the two-dimensional tame symbol as the commutator of a group-like monoidal groupoid which is obtained from some group of k-linear operators acting in a two-dimensional local…

### The Gersten conjecture for Milnor K-theory

- Mathematics
- 2008

We prove that the n-th Milnor K-group of an essentially smooth local ring over an infinite field coincides with the (n,n)-motivic cohomology of the ring. This implies Levine’s generalized Bloch–Kato…

### Tangent space to Milnor K-groups of rings

- Mathematics
- 2015

We prove that the tangent space to the (n + 1)th Milnor K-group of a ring R is isomorphic to the group of nth absolute Kähler differentials of R when the ring R contains 1/2 and has sufficiently many…

### On the Kernel and the Image of the Rigid Analytic Regulator in Positive Characteristic

- Mathematics
- 2009

We will formulate and prove a certain reciprocity law relating certain residues of the differential symbol dlog2 from the K2 of a Mumford curve to the rigid analytic regulator constructed by the…